On the complexity of optimal homotopies
Citation for published version (APA):Chambers, E. W., de Mesmay, A., & Ophelders, T. A. E. (2018). On the complexity of optimal homotopies. In A. Czumaj (Ed.), 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 (pp. 1121-1134). Association for Computing Machinery, Inc. https://doi.org/10.1137/1.9781611975031.73
DOI:
10.1137/1.9781611975031.73
Document status and date: Published: 07/01/2018
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On the complexity of optimal homotopies
Erin Wolf Chambers
∗Arnaud de Mesmay
†Tim Ophelders
‡Abstract
In this article, we provide new structural results and algo-rithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves γ1 and γ2 on
a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between γ1 and γ2
where the length of the longest intermediate curve is min-imized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quanti-tative homotopy theory to more practical applications such as similarity measures on meshes and graph searching prob-lems.
We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fr´echet distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting.
1 Introduction
This paper considers computational questions pertain-ing to homotopies: in broad terms, a homotopy between two curves in a topological space is a continuous defor-mation between these two curves. This can be formal-ized either in a continuous setting, where it constitutes one of the fundamental constructs of algebraic topol-ogy, but also in a more discrete one, where the input is a simplicial, or more generally cellular description of a topological space; this latter setting will be the focus of this article. While considerably more restrictive than the more traditional mathematical settings, this setting
∗Dept. of Computer Science, Saint Louis University.
email: echambe5@slu.edu
†Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab,
38000 Grenoble, France.
email arnaud.de.Mesmay@gipsa-lab.fr
‡Dept. of Mathematics and Computer Science, TU Eindhoven,
the Netherlands.
email: t.a.e.ophelders@tue.nl
is nonetheless of key importance in applications areas such as graphics or medical imaging, where inputs are generally represented by triangular meshes built upon scanned point sets from an underlying 3D object.
Investigating homotopies from a computational per-spective is a well-studied problem, dating back to the work of Dehn [13] on contractibility of curves, which has strong ties to geometric group theory. While deciding whether two curves in a 2-dimensional complex are ho-motopic is well-known to be undecidable in general (see for example Stillwell [29]), when the underlying space is a surface, efficient, linear-time algorithms have been designed to test homotopy [15, 17, 26]. In this article, we add a quantitative twist to this problem: the Ho-motopy Height problem consists, starting with two disjoint homotopic curves on a combinatorial surface, of finding the homotopy of minimal height, that is, where the length of the longest intermediate curve in the ho-motopy is minimized. (We refer the reader to Section 2 for formal definitions.) The notion of homotopy height has obvious appeal from a practical perspective, as it quantifies how long a curve has to be to overcome a hurdle: for example, deciding whether a bracelet is long enough to slide off over a hand without breaking is es-sentially the question of homotopy height. From a com-putational side, deformations of minimal height mini-mize the necessary stretch and can be used to quantify how similar curves are, as in map or trajectory analysis.
1.1 Our results
We begin by considering two curves forming the bound-ary of a discrete annulus, and study the homotopy be-tween these boundaries of minimal height. Our arti-cle leverages on recent results in Riemannian geome-try [10, 11], and in particular on a companion article co-authored with Gregory Chambers and Regina Rot-man [6] where we prove that in the RieRot-mannian setting, such an optimal homotopy can be assumed to be very well behaved. Firstly, it can be assumed to be an iso-topy, so that all the intermediate curves remain simple. Secondly, this isotopy can be assumed to only move in one direction and never sweeps any portion of the disk twice; we refer to this property as monotonicity, which we will define more precisely in Section 3.
These isotopy and monotonicity properties turn
γ0 γ1 γ0 γ1 γ0 γ1 γ0 γ1
Figure 1: Left: the height of a homotopy between homotopic curves γ0 and γ1 measures the maximum amount
an intermediate curve must stretch during the homotopy. Homotopies minimizing this amount of stretch measure the homotopy height. Right: the width of a homotopy measures the maximum length of a “slice” of the homotopy connecting the two boundary curves. Homotopies minimizing the length of this slice measure the homotopy width, also known as the homotopic Fr´echet distance.
out to be a key ingredient for computational purposes, once we translate those results to the discretized set-ting. First, via some surgery arguments, it allows us to prove that Homotopy Height is in NP (Theo-rem 5.1). The corresponding exponential time algo-rithm is to our knowledge the best exact algoalgo-rithm for Homotopy Height. We note that our setting is very general, and also implies NP-membership for a variant of Homotopy Height in a more restricted setting that was considered in earlier papers [3, 9, 22], as well as for Homotopic Fr´echet distance, where this was still open despite the recent articles investigating this dis-tance [7, 22]. Then, further surgery arguments allow us to provide an O(log n)-approximation algorithm for Homotopy Height (Corollary 6.2), by relying on an earlier O(log n) approximation-algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos [22] for homo-topy height in a more restricted setting. Finally, we show that monotonicity directly implies an equivalence between the Homotopy Height problem and a seem-ingly unrelated graph drawing problem which we call Minimal Height Linear Layout. Therefore, we ob-tain that this problem is also in NP and we provide an O(log n) approximation for it.
1.2 Related work
Optimal homotopies (for several definitions of optimal) have been studied extensively in the mathematical com-munity, mainly in Riemannian settings. This literature fits broadly in the setting of quantitative homotopy the-ory, initially introduced by Gromov [20], which aims at introducing a quantitative lens in the study of topo-logical invariants on manifolds. Probably the most ex-tensively considered notion of optimality is the study of homotopies minimizing the area swept; see for exam-ple [25] for an overview of some variants of this problem, or [30] for a discussion of how minimum area homo-topies and homologies are connected in higher dimen-sions. The notion of controlling the width of a homotopy
has also been studied [5, 23], and more recent work on minimal height homotopies [10, 11] laid the foundation for the results in this paper.
On the computational side, the rise of Fr´echet distance for measuring similarity between curves was a prime motivation for the notion of comparing two curves; see for example [1] for a survey. Generalizing the Fr´echet distance to curves on surfaces led to the homotopic Fr´echet distance, which is essentially the same as finding a minimum width homotopy given two input cycles on a surface. Polynomial time algorithms are known for the special case where the two input curves lie in the plane minus a set of obstacles [8]. Approximation algorithms exist for discrete settings where the two curves bound a disk [22].
More directly, minimum height homotopies have been studied from the computational perspective in various discretized settings [9, 22], although mainly to discuss the complexity of the problem. Indeed, as it was not known if the optimal height homotopy was even monotone, the complexity of the problem was completely open. Since the monotonicity result also holds in more geometric settings [6], a recent paper also examined one natural geometric setting, where the goal is to morph across a polygonal domain in Euclidean space with point obstacles; this work presents a lower bound that is linear in the number of obstacles, as well as a 2-approximation for the arbitrary weight obstacles and an exact polynomial time algorithm when all obstacles have unit weight [4]. The same problem also arises as a combinatorics question in lattice theory as a b-northward migration, where the authors leave monotonicity of such migrations as an open question [3].
1.3 Relations to graph searching and width parameters
This work also connects to sweep and search parameters in graph theory; see for example [18] for a survey of this topic. In each variant, the game consists of finding the
minimum number of searchers needed, where the goal is to find or isolate a hidden fugitive. For example, in the node searching variant, the fugitive hides on edges, all of which are originally contaminated, and the searchers clear an edge if two are on its incident vertices. In this variant, edges can be recontaminated if they are connected to a contaminated edge by a path without searchers, and the game ends when everything is decontaminated.
One key issue in these games is precisely that of monotonicity, or of determining whether in an opti-mal strategy, edges get recontaminated. In the node searching variant, monotonicity was established by La-paugh [24], and the argument was simplified by Bien-stock and Seymour [2]. One important corollary to monotonicity for these games is that it immediately shows the problem lies in NP, since a strategy can be certified by the list of edges cleared.
Our homotopy problem is quite similar to these graph parameters; sweeping a disk while keeping the length small is intuitively quite similar to blocking in a fugitive. While our problem does display minor technical differences with the aforementioned variant – most notably, our setting is naturally edge-weighted and the cost is measured on the edges and not the vertices – the key difference is the one of connectedness, as node-searching games may allow for disconnected strategies. An important variant of node searching, called connected node searching, requires additionally that the decontaminated space remains connected, but makes no restriction on the uncontaminated space.
For graph searching problems, the main argument to establish monotonicity does not maintain connec-tivity [2], and it was proven that an optimal strat-egy for connected node searching may indeed be non-monotone [31]. By contrast, Theorem 3.4 establishes monotonicity of the optimal homotopy in our setting, and the arguments differ radically from the ones of La-paugh and Bienstock and Seymour. As such, we identify in this paper a new variant of graph searching which is somewhat tractable (i.e., in NP) and introduce a new proof technique to establish monotonicity results.
Finally, when monotonicity is established, graph searching parameters are very intimately related to width parameters of graphs. Minimum cut linear ar-rangement (also known as cut-width) is closely con-nected to the Minimum Height Linear Layout prob-lem, which we show to be equivalent to Homotopy Height, but the key difference is that it may break the embedding of the graph. Thus, NP-hardness re-ductions for this problem [27] do not imply hardness for our problem. Connected variants of various width parameters give rise to connected pathwidth [14] and
connected treewidth [19], but in contrast to our homo-topies, these parameters are only connected “on one side”, which makes them incomparable. We believe that the “doubly-connected” aspect of homotopy height makes it a worthwhile new graph parameter which could offer insights to other parameters in this area.
Outline of the paper. After introducing the pre-liminaries in Section 2, we lay the foundations of this work by explaining the structural theorems we rely on in Section 3. In Section 4 we establish surgery lemmas based on the idea of retractions. Then, in Section 5 we prove that Homotopy Height is in NP. In Section 6 we draw connections with Homotopic Fr´echet Dis-tance, and we leverage on these connections to pro-vide an O(log n)-approximation algorithm for Homo-topy Height.
2 Preliminaries
Homotopy and Isotopy. Let Σ be a surface, endowed with a cellularly embedded graph G with n vertices such as in Figure 2, and let γ0 and γ1 be two
simple cycles on G bounding an annulus.
1 1 1 1 1 10 20 30 20 1 1 1 γ0 γ1
Figure 2: Example instance G, based on an example in [3].
A discrete homotopy h between γ0 and γ1 is a
sequence of cycles h(ti) with 0 = t0 ≤ · · · ≤ ti ≤
· · · ≤ tm = 1, with h(t0) = γ0 and h(t1) = γ1 and
any two consecutive paths h(ti) and h(ti+1) are one
move apart. The intermediate curves h(t) are called
level curves or intermediate curves. A move is either a face-flip, an edge-spike or an edge-unspike (flip, spike or unspike, for short). A face-flip for a face F replaces a single subpath p of h(ti) ∩ ∂F with the
path ∂F \p in h(ti+1). Anedge-spikefor an edge u → v
replaces a single occurrence of a vertex u ∈ h(ti) by the
path u → v → u consisting of two mirrored copies of that edge in h(ti+1). Symmetrically, anedge-unspike
replaces a path u → v → u of h(ti) by the single
vertex u in h(ti+1). Thelength`(h(i)) of a path h(i) is
the sum of the weights of its edges (with multiplicity). The height of a homotopy h is the length of the longest path h(ti). Anoptimal homotopy is one that
γ0
3 35 24 24 5 12
γ1
h(t5)
h(t0) h(t1) h(t2) h(t3) h(t4)
Figure 3: An optimal homotopy h of height 35 for the instance of Figure 2. minimizes the height. Thehomotopy height between
γ0and γ1is the height of an optimal homotopy between
γ0 and γ1. Figure 3 illustrates an optimal homotopy
that uses only face-flips for the instance of Figure 2. Cross Metric Surfaces. For most purposes, it is more convenient to think of this discrete model in a dual way, relying on the cross-metric surfaces [12] which are becoming increasingly used in the computational geometry and topology literature. In this dual setting, a cross-metric surface is a surface Σ endowed with a weighted (dual) graph G∗.
Assuming the primal surface is connected, we ob-tain this dual graph by gluing a disk to each boundary component, taking the dual graph, and puncturing the vertices corresponding to the added disks, without re-moving the adjacent edges. Such that these (dual) edges end at the boundary of the cross-metric surface instead of at a vertex, see Figure 4.
For a curve γ on Σ with a finite number of crossings with G∗, its length `(γ) is the weighted sum of the
crossings γ ∩ G∗. Now, a homotopy between γ
0 and γ1
is a homotopy in the usual sense, that is, a continuous map h : S1× [0, 1] → Σ such that h(·, 0) = γ
0 and
h(·, 1) = γ1, except that we require that the values
of t for which h(·, t) is not in general position with G∗ are isolated, and each such curve has at most one such degeneracy1 h(x, t) with G∗. As before, the height
of a homotopy is defined as the maximal length of an intermediate non-degenerate level curve h(t). A
1Any homotopy can be made so by a small perturbation
with-out increasing the height, so we always consider this hypothesis fulfilled in the remainder of the article.
γ0
γ1
Figure 4: Dual representation of Figure 2.
homotopy is an isotopy if all the intermediate curves are simple.
Given a homotopy h∗ in this setting, we obtain a
discrete homotopy h on the primal graph G on Σ as follows. Pick a curve h∗(t
i) in each maximal interval of
non-degenerate curves in h∗ (all curves in such interval
have the same crossing pattern with G∗, and therefore
the same length). Let h(ti) be the curve on G whose
sequence of vertices and edges corresponds to the se-quence of faces and edges of G∗ visited by h∗(ti). This
model is dual to the previous one, and Figure 5 illus-trates how any move (flip, spike or unspike) connects two intermediate curves h(ti) and h(ti+1). We say a
discrete homotopy is an isotopy if it can be obtained from an isotopy in the dual setting.
3 Isotopies and monotonicity of optimal homotopies
We begin by restating and explaining the two structural results that we will rely on. Introducing the relevant Riemannian background lies outside of the scope of this paper, so we will simply advise the uninitiated reader to picture a Riemannian surface as a surface embedded into R3, where the metric on the surface is the one
induced by the usual Euclidean metric of R3. Thanks
to the Nash-Kuiper embedding theorem (see [21]), this naive idea looses no generality. We refer to standard textbooks on the subject for more proper background on Riemannian geometry, for example do Carmo [16].
The first theorem shows that up to an arbitrarily small additive factor, the homotopy of minimal height between two simple closed curves can be assumed to be an isotopy.
spike flip unspike
Figure 5: Three moves in the primal (left) and dual (right) representation.
Theorem 3.1. ([10, Theorem 1.1]) Let Σ be two-dimensional Riemannian manifold with or without boundary, and letγ0andγ1be two non-contractible
sim-ple closed curves which are homotopic through curves bounded in length byL via a homotopy γ. Then for any ε > 0, there is an isotopyeγ from γ0toγ1through curves of length at most L + ε.
Remark. The non-contractibility hypothesis is re-quired because if M is not a sphere, contractible cy-cles with opposite orientations are homotopic but not isotopic. However, if we disregard the orientations, the result holds in full generality.
This theorem has the following discrete analogue: Theorem 3.2. Let (Σ, G∗) be a cross-metric surface, and let γ0 and γ1 be two non-contractible simple closed
curves on (Σ, G∗) which are homotopic through curves
bounded in length by L via a homotopy γ. Then there is an isotopy eγ from γ0 toγ1 through curves of length
at most L.
The proof is exactly the same as the one of Theo-rem 3.1, except that it does not need the ε-slack: this was required to slightly perturb the curves so that they are simple but in the discrete setting it can be done with no overhead.
The second theorem shows that, when the starting and finishing curves of a homotopy are the boundaries of the manifold, there exists an optimal homotopy that is monotone, i.e., that never backtracks, once again up to an arbitrarily small additive factor. Formally, if γ is an isotopy (which we can assume the optimal homotopy to be, by Theorem 3.1) between γ0 and γ1, for 0 ≤ t ≤ 1,
the curves γt and γ1 bound an annulus At. Then the
isotopy γ ismonotoneif for t < t0 < 1, γt0 is contained
in At.
Theorem 3.3. ([6]) Let M be a Riemannian annulus with boundaries γ0 and γ1 such that there exists a
homotopy betweenγ0andγ1of height less thanL. Then
there exists a monotone homotopy betweenγ0andγ1of
height less thanL.
Note that the ε-slack of Theorem 3.1 is also present here but is hidden in the open upper bound on the height. In this theorem, as was observed by Chambers and Rotman [11], crediting Liokumovitch, the hypothe-sis that the manifold is entirely comprised between both curves is necessary: see [11, Figure 5] for a counter-example.
In the discrete setting, the corresponding result is the following, where the definition of monotonicity is the same:
Theorem 3.4. Let (Σ, G∗) be a cross-metric annulus with boundaries γ0 and γ1 such that there exists a
homotopy between γ0 and γ1 of height L. Then there
exists a monotone isotopy between γ0 and γ1 of height
L.
The proof is exactly identical to the one in the Riemannian settting and it yields a slightly stronger result, since the cross-metric setting removes the need for perturbations and thus the need of an ε-slack. Remark. Let us observe that the discrete theorems are in some way more general than the Riemannian ones: not only do they bypass the need for some ε-slack, but they also directly imply their Riemannian converses by the following reduction. Starting with a Riemannian surface, and a (non-monotone) isotopy between two disjoint curves, one can find a triangulation of the surface allowing, at an ε-cost, to approximate the isotopy using only elementary moves. Then, after making this isotopy monotone in the discrete setting, one can translate it back into a monotone isotopy in the Riemannian setting by interpolating between the face and edge moves.
4 Retractions and pausing at short cycles In this section, we establish several technical lemmas which are necessary for our proofs in the next section. For simple closed curves β and γ bounding an annulus, denote that annulus by A(β, γ). Let S(β, γ) be the set of closed curves in A(β, γ) homotopic to boundaries β and γ, that do not intersect homotopic curves of shorter length. Then, for any point p ∈ α ∈ S(β, γ), α is a shortest closed path through p in its homotopy class. Let G(β, γ) be the set of minimum length simple closed curves homotopic to the boundaries of A(β, γ), then G(β, γ) ⊆ S(β, γ).
We now introduce the concept of a retraction of a homotopy, which gives a way to shortcut a homotopy at a given curve, provided it is a curve of S(β, γ). This idea is implicit in Chambers and Rotman [11, Proof of Theorem 0.7], and we refer to their article for more details. For a monotone isotopy h between boundaries of an annulus A, and a homotopic annulus A0 ⊂ A,
define the retraction h|A0
(t) of h(t) to A0 as the
same curve with each arc of h(t) \ A0 replaced by the
shortest homotopic path along the boundary of A0.
Although paths along ∂A0 (dis)appear discontinuously
as t varies, h|A0
can be obtained in the form of a discrete homotopy by (un)spiking these paths as they (dis)appear. The resulting homotopy h|A0is a monotone
isotopy.
Lemma 4.1. If α ∈ S(α, γ) and A(α, γ) ⊆ A(β, γ), and h is a monotone isotopy from β to γ of height L,
then h|A(β,α) is a monotone isotopy from β to α with
height at mostL.
Proof. The retraction h0 = h|A(β,α) is a monotone
isotopy from h0(0) = β to h0(1) = α. Let t0 be the
maximum t for which h(t) intersects A(β, α). For t ≥ t0,
we have h0(t) = α and therefore |h0(t)| = |α| ≤ |h(t0)| ≤
L. For t ≤ t0, each arc a of h(t) \ A(β, α) is replaced
in h0(t) by a homotopic path b along α with |b| ≤ |a|, and thus |h0(t)| ≤ |h(t)| ≤ L. Hence height(h0) ≤ L.
Lemma 4.2. If α ∈ S(β, γ), and h is a monotone isotopy fromβ to γ of height L, then there is a monotone isotopy from β to γ of height at most L having α as a level curve.
Proof. We have α ∈ S(α, β) and α ∈ S(α, γ). So by Lemma 4.1, the monotone isotopies h|A(β,α)from β to α
and h|A(α,γ)from α to γ have height at most L and can
be composed to obtain a monotone isotopy from β to γ of height at most L with α as a level curve.
Lemma 4.3. Let Π = {π1, . . . , πm} be a set of paths
from γ0 to γ1 without proper pairwise intersections,
where each πi is a shortest homotopic path inA(γ0, γ1)
between its endpoints. If h is a monotone isotopy fromγ0 toγ1 of heightL, then there exists a monotone
isotopy of height at mostL whose level curves all cross eachπi at most once (after infinitesimal perturbations).
Proof. Denote by c(a, b) the number of proper intersec-tions of curves a and b, and by cΠ(a) = Pπ∈Πc(a, π)
the total number of intersections of a with Π. Let Ch=
maxtcΠ(h(t)) be the maximum total number of
inter-sections over all t, and let Ih be the set of maximal
intervals (τ0, τ1) with cΠ(h(t)) = Ch if t ∈ (τ0, τ1) ∈ Ih.
If Ch= m, each level curve of h crosses each πi exactly
once and we are done, thus we assume in the following that cΠ(h(0)) = cΠ(h(1)) = m < Ch.
If Ch > m, we obtain a homotopy h0 from h
with Ch0 < Ch by, for each interval (τ0, τ1) ∈ Ih,
replacing subhomotopy h|(τ0,τ1) of h by some h
∗ =
h0|
(τ0,τ1) with Ch∗< Ch.
Consider a single interval (τ0, τ1) ∈ Ih and let A =
A(h(τ0), h(τ1)). Then Π ∩ A consists of Ch subarcs
of Π, each connecting the two boundaries of A. For t ∈ (τ0, τ1), h(t) intersects each such arc exactly once, and
each h(t) intersects these arcs in the same order. Among the components of A \ Π, there is a disk D0bounded by
one arc of h(τ0) and two arcs of πi∩ A, and a disk D1
bounded by one arc of h(τ1) and one arc of πj, such that
these disks contain no other arcs of Π.
We can find α ∈ G(h(τ0), h(τ1)) that intersects
any arc of A ∩ Π at most once (in the same order
π1 π2 π3 γ0 γ1 h(τ0) h(τ1) α D0 D1
Figure 6: Choosing α such that Ch∗< Ch.
as h(t)), and does not intersect the interiors of D0
and D1 (because the two arcs of Π on their
bound-ary form a shortest path). Then cΠ(α) < Ch and the
retraction h0 = h|A(h(τ0),α) has Ch0 < Ch, since any
arc h0(t) has fewer intersections than h(t) has with Π
(in particular with the boundary of D1). Symmetrically,
for h1= h|A(α,h(τ1)) we have Ch1 < Ch. Since the
com-position h∗ = h0h1 is a homotopy from h(τ0) to h(τ1)
with Ch∗< Ch and height at most L (by Lemma 4.2),
we can use this as a replacement for h|(τ0,τ1)in h
0. By
induction, we obtain a homotopy of height at most L whose level curves all cross each πi at most once.
5 Computing homotopy height in NP
In this section, we show that in the discrete setting, there is an optimal homotopy with a polynomial number of moves. First, we show that there is a homotopy that flips each face exactly once.
Lemma 5.1. For an annulus (Σ, G) bounded by γ0
andγ1, there is a homotopy of minimum height between
γ0 andγ1 that flips each face of G exactly once.
Proof. By Theorem 3.4, some homotopy h of minimum height is a monotone isotopy. For two consecutive level curves h(t) and h(t0) in a monotone isotopy, the move between h(t) and h(t0) flips face F if and only if F lies in A(h(t0), γ1) or A(h(t), γ1) but not both.
Be-cause A(h(0), γ1) contains all faces, and A(h(1), γ1)
con-tains none, each face is flipped at least once. By mono-tonicity, we have for 0 ≤ t0 < t ≤ 1, that A(h(t0), γ
1) ⊇
A(h(t), γ1). So, if face F does not lie in A(h(t), γ1), it
will not be flipped again in h|(t,1]. Hence each face is
flipped exactly once.
It remains to show that each edge is involved in a polynomial number of (un)spike moves; note that this does not directly follow from monotonicity, since a second spike of the same edge does not violate monotonicity (as can easily be seen in the dual setting).
a
b
c
d
Figure 7: Delaying spikes (a). Canceling spikes with unspikes (b) or faces (c). Part of a reduced isotopy (d).
Postponing spikes. Before we bound the number of spike moves, we transform an optimal monotone isotopy h into one where each spike move is delayed as much as possible, and each unspike move happens as soon as possible. We explain this transformation in the dual setting.
Suppose a spike move occurs for edge e be-tween h(ti) and h(ti+1), then denote by s the (unique)
arc of A(h(ti), h(ti+1)) ∩ G∗ both of whose endpoints lie
on h(ti+1). This arc is a subarc of the dual edge e∗.
Con-sider the maximum j > i, for which the component sj
of e∗ ∩ A(γ
0, h(tj)) containing s has both endpoints
on h(tj), and for all ti < t ≤ tj, curve h(t) has exactly
two crossings with sj (so the only action performed on
arc sjwas the spike between h(ti) and h(ti+1)). Then sj
and h(tj) enclose a disk Dj. If the interior of Dj
con-tains no edges of G∗, we can delay the spike of e at least until just before tj, as illustrated in Figure 7 (a),
where Dj is shaded.
Depending on what happens in the move be-tween h(tj) and h(tj+1), we may transform the isotopy
further. This move is either (1) an unspike attached to sj, or (2) a face-flip connected to one endpoint or
(3) both endpoints2 of s
j, or (4) a face-flip or spike
in-side Dj+1. In cases (1) and (2), we cancel the spike
against the unspike or flip, as illustrated in Figure 7 (b) and (c). We do not postpone the spike in cases (3) and (4). Symmetrically, unspike moves can be made to happen earlier. Observe that these operations cannot increase the height of a homotopy since each level curve in the resulting homotopy crosses a subset of the edges of some curve in the original homotopy.
Call a homotopy reduced if it is the result of applying the above rules to h until no spike can be canceled or postponed until after a flip or unspike, and no unspike can be canceled or be made to happen before any prior flip or spike. Starting from an optimal monotone isotopy, the reduced isotopy is also an optimal monotone isotopy. Lemma 5.2 captures a structural property of reduced homotopies.
2This happens only if the primal edge is adjacent to only one
face of G.
Lemma 5.2. Between any two consecutive face-flips in a reduced isotopy lies a single (possibly empty) path of unspike moves followed by a (possibly empty) path of spiked moves.
Proof. In a reduced homotopy, no unspike follows a spike move, and any spikes that remain ‘surround’ the next face-flip (if any), see Figure 7 (d). Symmetrically, all unspikes between two consecutive face-flips surround the previous face-flip (if any). From the primal perspec-tive, these unspike moves form a path from the previ-ously flipped face and spike moves form a path towards the next flipped face.
Any reduced homotopy starts with zero or more unspikes from γ0, after which a possibly empty path
of spikes to the first face-flip occurs, then that face is flipped, and a possibly empty path of unspikes enabled by this flip occurs. Subsequently, a spiked path, face-flip, and unspiked path occur for the remaining faces. Finally, a sequence of spikes towards γ1 may
occur. We may assume that on γ0 and γ1, any two
consecutive edges are different, such that no immediate unspike moves are possible from γ0, and no immediate
spike moves are possible to γ1. Otherwise we may by
Lemma 4.1 perform those moves immediately without increasing the homotopy height.
Bounding spike moves. We are now ready to bound the number of spike and unspike moves in an optimal homotopy. Call a homotopy h good if it is a minimum-height reduced monotone isotopy and it has a minimum number of moves. By Theorems 3.2 and 3.4, the height of h is the homotopy height between γ0
and γ1.
Define an edge-spike of an edge e to be between
existing copies of e, if the portion of the dual edge e∗ crossed by the (dual) level curve, lies between two existing crossings of the level curve with e∗, such as in Figure 8. We show that such spikes never appear in h.
Lemma 5.3. If homotopy h is good, there are no spikes between existing copies of any edge e.
u e v
Figure 8: Development of a spike between existing copies of e. Part of the graph in red (dual) and blue (primal) and the level curve in gray dashed (dual) and black (primal, perturbed).
p1 p4 p1 p4 p2 p3 p1 p4 p1 p2 p1 p4 p1 p2 p1 p4 p2 p3 p1 p4 p1 p4 p2 p3 p1 p4 C1 C3 C2 e∗
Figure 9: Top: the neighborhood of e∗ throughout h. Bottom: the reconnected homotopy, reducing crossings
with e∗. From left to right: the homotopy just before τ
0, just after τ0, between τ0and τ1, just before τ1, and just
after τ1.
Proof. Suppose the move from h(ti) to h(ti+1) is the
last move between existing copies of the same edge, and assume this move is a spike of edge e = (u, v) from u to v. In the dual setting, consider the component π of e∗∩ A(h(t
i), γ1) that is crossed by the spike move
(highlighted in Figure 8). Let c(t) be the number of crossings of h(t) with π, then for some τ0 between ti
and ti+1, c(τ0) = 3, and for some unique τ1> τ0, c(τ1) =
3 again, and for τ0< t < τ1, we have c(t) = 4 (because
we assumed this was the last spike between existing copies of any edge).
For τ0 < t < τ1, label the four crossings of h(t)
with π by p1(t), p2(t), p3(t), and p4(t), in order along e∗,
so the spike move at τ0creates p2 and p3. Consider the
three components C1(t), C2(t) and C3(t) of A(h(t), γ1)\
π, such that C1 touches p1 and p2 from the dual face
of u, and C2 touches p3 and p4 from the dual face of u,
and C3 touches e∗ in two segments from the dual face
of v. Because C3 lies between C1 and C2, h will first
contract either component C1 or C2, namely at h(τ1).
Assume without loss of generality that C2 contracts
first.
We modify h|[τ0,τ1] such that any level curve
crosses π at most twice by reconnecting the
neighbor-hood of π, whose local structure evolves exactly as de-picted in the top row of Figure 9. We essentially re-move crossings p2 and p3, and reconnect ∂C1(t) ∩ h(t)
with ∂C2(t)∩h(t) using a (zero-length) segment along π
in face u∗. On the other side, consider the arc of ∂C 3(t)∩
h(t) ∩ v∗ with p4(t) as endpoint. We cut this arc in two
subarcs a and b, where a has p4(t) as endpoint, and
connect the other endpoint to the arc of ∂C3(t) ∩ h(t)
at the endpoint at p2(t) using a segment along π in v∗.
Similarly, we connect the endpoint of that at p3(t) to
the loose end of b. These reconnections are depicted in the bottom row of Figure 9. A more global view (cor-responding to Figure 8) is illustrated in Figure 10.
Observe that the reconnected curves can be made to appear continuously in such a way that they form a monotone isotopy. Because level curves only changed in the neighborhood of π, where they were shortened by avoiding the crossings with π, we have an isotopy whose height is at most that of h, and in which at least one spike is removed. So, because h was optimal, we have constructed an optimal monotone isotopy with fewer moves. Therefore, the corresponding reduced isotopy also has fewer moves, contradicting that h was good.
Figure 10: Figure 8 after a local surgery that avoids the spike between copies of e. s1 s2 s3 σ0 σ1 σ2 σ3
Figure 11: A local surgery to avoid five spikes of the same edge on a single spiked path.
Our final step towards bounding the number of edge spikes is to derive a contradiction if for some interval [τ0, τ1] without face-flips, an edge e is spiked
(or unspiked) 5 times in h|[τ0,τ1]. The proof is similar to
that of Lemma 5.3.
Lemma 5.4. For a good homotopy h, any subhomo-topy h|[τ0,τ1] contains either a face-flip, or at most 4
spike (and at most 4 unspike) moves of the same edge. Proof. Suppose h|[τ0,τ1] contains no face-flip, then
be-cause h is reduced, the spike moves in h|[τ0,τ1] form a
path σ of spike moves in G. Assume for a contradiction that some edge e = (u, v) lies on σ at least 5 times. We say two spikes s1and s2are consecutive on e∗if no spike
occurs on the arc of e∗ between the first crossing of s 1
with e∗ and the first crossing of s
2 with e∗.
Because by Lemma 5.3, h does not contain spikes between existing copies of edges, we can find three spikes s1, s2 and s3 of e on σ where s1 and s2 as well
as s2 and s3are consecutive on e∗, and s1happens
be-fore s2 and s2happens before s3. Let σ0, σ1, σ2and σ3
be the subpaths of σ such that σ = σ0s1σ1s2σ2s3σ3,
also labeled in Figure 11.
To get rid of spike s2, we connect σ0s1σ1to σ2s3σ3
in an alternative way. Figure 12 illustrates all possible ways s1, s2and s3(in the dotted area) can be connected
by σ, and how our method will reconnect σ without s2.
Formally, to decide where this reconnection takes place, we consider the components of A(h(τ1), γ1) \ π, where π
is the arc of e∗between its intersections with s1and s3.
There are three components, component C1 touching π
and σ1, component C2 touching π and σ2, and
compo-nent C3touching σ entirely, and touching π in two arcs.
The component that h contracts first is either C1or C2
(since C3lies between the other two).
First consider the case where C1 is contracted first,
then the path σ2s3σ3 starts in the dual face of the
endpoint of s2. Note that there is a (zero-length) path
between the start or endpoint of s1 and the endpoint
of s2 because s1and s2 are adjacent along e∗. Use this
zero-length path to connect σ2s3σ3 to σ0s1σ1 at the
start or endpoint of s1 and call the resulting tree λ.
Figure 12: Cases for shortcutting spiked paths visiting the same edge often. The neighborhood of the repeated edge is dotted and the component contracted first is shaded.
We claim we obtain an optimal monotone isotopy h0 from h by replacing the spiked path σ by the spiked tree λ, and removing the unspike move of e∗ following the contraction of C1. Up until the creation of λ, the
move sequence is the same as in h. Since λ contains a subset (all spikes except s2) of the spikes of σ, the spiked
tree can be created without surpassing the height of h. After the creation of σ in h and λ in h0, locally, the level
curves of h and h0 differ only in a small neighborhood
of π, so that all moves of h except those crossing π can still be performed in h0. Because s
2 is the only spike
along e∗ that lies between s
1 and s3, the next move
that crosses π is the unspike move, call it z, following the contraction of C1. The level curve of h0 just before z
is identical to the level curve of h just after z, so it is safe to omit move z in h0. All subsequent level curves of h and h0 are identical, so we conclude that h0 is an optimal monotone isotopy (with fewer moves). Therefore, the reduced monotone isotopy of h0has fewer
moves, contradicting that h was good.
The proof for the case where C2 contracts first, is
symmetrical, except that the spiked tree λ is created differently. In this case, we define λ to be σ0s1σ1, whose
endpoint is connected to σ2s3σ3at the start or endpoint
of s3. When spiking this tree, the direction of the spikes
on σ2 (and sometimes σ3) is reversed, but this does not
affect the proof.
Hence, in a good homotopy, no spiked path spikes
the same edge 5 times.
Theorem 5.1. For γ0andγ1bounding an annulus with
n faces and m edges, there is a homotopy of minimum height that has at most O(mn) moves. Therefore, de-ciding whether their Homotopy Height is at most L is in NP.
Proof. Let n be the number of faces, and m the number of edges in G. As a direct consequence of Lemmas 5.1 and 5.4, there is a good homotopy that spikes each edge at most 4(n + 1) times and unspikes each edge at most 4(n + 1) times. So there is a homotopy of minimum height that has at most 8m(n + 1) + n = O(mn) moves. Testing whether this homotopy indeed has height at most L can be done by computing the maximum length over its (polynomially many) level curves, each containing a polynomial number of edges, and comparing this maximum with L. Given a good homotopy, all of this can be done in polynomial time assuming addition and comparisons of numbers takes polynomial time.
We note that the Homotopy Height problem can also be defined in slightly different settings, for example • γ0 and γ1 are two paths with common endpoints s
and t, such that γ0∪ γ1 is the boundary of a
fixed endpoints, and we are interested in comput-ing the optimal height of this homotopy. This is the Homotopy Height problem considered by E. Chambers and Letscher [9].
• There is a single curve γ forming the boundary of a combinatorial disk. This curve is contractible, and we are interested in computing the optimal height of such a contraction. This is one of the settings considered in [6].
In both these cases, the Theorems 3.2 and 3.4 have analogues establishing that some optimal homotopy is an isotopy and is monotone. The rest of our proof tech-niques then readily apply, and prove that the Homo-topy Height problem in these cases is also in NP. The next section investigates more distant variants.
6 Variants and approximation algorithms 6.1 Homotopic Fr´echet distance
There is a strong connection between the problem of Homotopy Height and the problem of Homotopic Fr´echet distance, which we now recall. As in [22], our setting is the one of a disk D with four points p0, q0,
q1 and p1 on the boundary, connected by four disjoint
boundary arcs γ0, γ1, P and Q, with γ0from p0to q0; γ1
from p1 to q1; P from p0 to p1; and Q from q0 to q1,
see Figure 13, left. A homotopy between γ0 and γ1
is a series of elementary moves connecting curves of D with one endpoint on P and the other on Q, where the collection of curves starts at γ0 and ends at γ1. The
Homotopic Fr´echet distance between P and Q is the height of a homotopy between γ0 and γ1of minimal
height. The common intuition for this distance is that it is the minimal length of a leash needed for a man on P to walk his dog along Q, where the leash may stretch but cannot be lifted out of the underlying space.
We note that this is slightly different than the original setting for homotopic Fr´echet distance in the original work [8], where an exact algorithm is presented for the plane minus a set of polygonal obstacles. In the original work, the start and end leashes are not fixed, and in fact the bulk of the work is in determining an optimal relative homotopy class in order to find the best homotopy.
Proposition 6.1. The Homotopic Fr´echet dis-tance problem is in NP.
Proof. We reduce Homotopic Fr´echet Distance to Homotopy Height using the following construction. We add a vertex v and edges of weight K between this vertex and all the vertices of the paths P and Q, where K is a constant greater than the sum of the weights
of the edges of the disk, as well as all the intermediate triangles, see Figure 13, right. This results in a pinched annulus A, with two boundaries γ0
0 and γ10 obtained
from the paths γ0 and γ1, both completed into closed
curves using the additional vertex v. We claim that an optimal homotopy between γ0and γ1translates into an
optimal homotopy in A between γ00 and γ10, and
vice-versa. Indeed, by Lemma 4.3, there exists an optimal homotopy in A such that any intermediate curve crosses the shortest path between γ00 and γ10 exactly once, and
in our case the shortest path is the zero length path starting and ending at the vertex v. Furthermore, if the weight K is big enough, the level curves of an optimal homotopy between γ0
0 and γ10 will always use exactly
two of the edges of weight K, since two are needed but any more would be too expensive. Thus, an optimal homotopy between γ0
0and γ10 translates directly into an
optimal homotopy between γ0 and γ1 after cutting on
v and removing the edges linked to v and vice-versa. The homotopy height is increased by exactly 2K in this translation.
Har-Peled, Nayyeri, Salvatipour and Sidiropoulos [22] provide an algorithm to compute in O(n log n) time a homotopy of height O(d log n), where d is a lower bound on the height of an optimal homotopy, and n is the complexity of Σ. In particular, one can set d to be the maximum of kγ0k, kγ1k, the diameter of Σ, and half of
the total weight of the boundary of any face. This yields an O(log n) approximation for Homotopic Fr´echet distance3. We show here that their algorithm can be adapted to yield an O(log n) approximation for Homotopy Height.
Proposition 6.2. One can compute in O(n log n) time an O(log n)-approximation of Homotopy Height. Proof. Starting with an annulus and two boundary curves γ0 and γ1, we first compute a shortest path P
between the boundary curves γ0 and γ1 and cut along
P to obtain a disk D. This brings us to the setting of Homotopic Fr´echet Distance, and we can apply the aforementioned algorithm and obtain a homotopy h. In order to recover a homotopy between γ0 and γ1,
we glue back the disk along P into an annulus, and the level curves of h are completed into closed curves by using subpaths of P, this gives us a homotopy h0. It remains to show that this is an O(log n) approximation of the optimal homotopy. By Lemma 4.3, some optimal homotopy between γ0 and γ1 has level curves cutting
3This algorithm assumes triangular faces, but using our
defi-nition of d, one can extend the algorithm of [22] to also work with polygonal faces.
Σ γ0 γ1 P Q p0 p1 q1 q0 Σ γ0 γ1 P Q p0 p1 q1 q0 K v K
Figure 13: The setting of homotopic Fr´echet distance.
P exactly once. Thus, the height L of an optimal homotopy in the disk D is a lower bound for the height of an optimal homotopy in the annulus A. Furthermore, each level curve γt of h0 consist of two subpaths, one
being a level curve h(t) of h and the other being a subpath Pt0 of P. Since P is a shortest path, Pt0 is
also a shortest path between its endpoints, so it is shorter than h(t) since they have the same endpoints. By construction, the length of h(t) is O(L log n), and thus the length of γt is O(2L log n) = O(L log n). This
concludes the proof.
6.2 Minimum height linear layouts
We also show that a seemingly unrelated graph draw-ing problem is directly equivalent to the Homotopy Height problem. A linear layout is an embedding of a planar graph where the edges have isolated tangencies with the vertical line, and all the vertices have distinct x coordinates. The Minimum Height Linear Layout problem is the following one: Given a planar embedding of an edge-weighted graph G, find a homeomorphic lin-ear layout of G in R2 such that the maximal weight of
the vertical lines is minimized. Here, the weight of a vertical line is the sum of the weights of the edges that it crosses, and (similarly to the cross-metric setting), vertical lines crossing tangent to the edges or crossing vertices are not counted.
Theorem 6.1. The Minimum Height Linear Lay-out problem is equivalent to the Homotopy Height problem.
Proof. Indeed, a linear layout of a planar graph G naturally induces a discrete homotopy sweeping its dual graph G∗. More formally, we drill a small hole around
the vertex dual to the outer face of G, and we view its complement as a disk D which is a cross-metric surface for the graph G. Since the hole was drilled in the middle of the face of G, its boundary has zero
length. We pick two arbitrary vertices s and t on it, which cuts the boundary into two paths L and R. Then we claim that a minimum height linear layout of G is equivalent to a homotopy of minimum height between L and R (where the endpoints are fixed)4. Indeed,
whenever the sweep of R2 induced by the vertical
lines crosses an edge or passes a vertex, by the dual interpretation of homotopies with cross-metric surfaces outlined in the preliminaries, it amounts to doing a face or an edge move, and thus the whole vertical sweep defines a homotopy between the two paths L and R. Furthermore, this homotopy is an isotopy, since the vertical lines are simple, and a monotone one since they only go in a single direction. Conversely, a discrete homotopy of optimal height between L and R can be “straightened” into a linear layout: by Theorem 3.4, one can assume such a homotopy h to be an isotopy and to be monotone, and therefore the succession of dual moves of h with respect to G are homeomorphic to a sweep of G by vertical lines, as pictured in Figure 14. An optimal homotopy amounts, via this homeomorphism, to finding a linear layout of minimal weight.
In particular, the Minimum Height Linear Lay-out problem is in NP and admits an O(log n)-approximation algorithm.
4The somewhat artificial construction with L and R forces the
homotopy to go through the outer face of G at all times.
3 35 24 24 5 12 30
20 20 10
Figure 14: Dual representation of Figure 2 (left) and Figure 3 (right).
Acknowledgements. We are grateful to Tasos Sidiropoulos for his involvement in the early stages of this research, and to Gregory Chambers and Regina Rotman for helpful discussions. This research began while partially supported through the program “Si-mons Visiting Professorship” by the Mathematisches Forschungsinstitut Oberwolfach in 2015. Erin Cham-bers is supported in part by NSF grants IIS-1319944, CCF-1054779, and CCF-1614562. Arnaud de Mesmay is partially supported by the ANR project ANR-16-CE40-0009-01 (GATO). Tim Ophelders is supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.023.208.
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